In the field of calculus of variations in mathematics, the method of Lagrange multipliers on Banach spaces can be used to solve certain infinite-dimensional constrained optimization problems.
Let U be an open subset of X and let f : U → R be a continuously differentiable function.
Then there exists a Lagrange multiplier λ : Y → R in Y∗, the dual space to Y, such that Since Df(u0) is an element of the dual space X∗, equation (L) can also be written as where (Dg(u0))∗(λ) is the pullback of λ by Dg(u0), i.e. the action of the adjoint map (Dg(u0))∗ on λ, as defined by In the case that X and Y are both finite-dimensional (i.e. linearly isomorphic to Rm and Rn for some natural numbers m and n) then writing out equation (L) in matrix form shows that λ is the usual Lagrange multiplier vector; in the case n = 1, λ is the usual Lagrange multiplier, a real number.
In terms of the above theorem, the constraint g would be given by However this problem can be solved as in the finite dimensional case since the Lagrange multiplier
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